ipbuker_graph07İİTTÜÜ--SUNY SUNY 20042004--2005 2005 FallFall

Graphic Graphic CommunicationsCommunications

Lecture 8: Projections

Assoc. Prof.Dr. Cengizhan Assoc. Prof.Dr. Cengizhan İİpbpbüükerker

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Projections

• The projections used to display 3D objects in 2D are called Planar Geometric Projections

• For computer graphics, the main types of projection used are:– Perspective Projections– Parallel Projections

• Perspective projections are defined by a Center Of Projection (COP) and a projection plane

• Parallel projections are defined by a Direction Of Projection and a projection plane

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Perspective Projections• Perspective projections are defined by a Center Of

Projection (COP) and a projection plane

• Perspective projections produce a perspective foreshortening effect

• They tend, therefore, to appear more realistic thanparallel projections (this is how our eye and a camera lens form images)

• Object positions are transformed to the view plane along lines that converge to a point (the COP)

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Perspective projections

Projection lines

(or centre of projection)

The projected view of an object is determined by calculating the intersections of the projection lines with the view plane

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Perspective ProjectionsParallel lines in the 3D model which are not parallel to the projection plane, converge to a vanishing pointIf the vanishing point lies on a principle axis, it is called a principle vanishing point

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Perspective Projections - Cont.• The number of principal vanishing points is

determined by the number of principal axes cut by the projection plane.

• If the plane only cuts the z axis (most common), there is only 1 vanishing point.

• 2-points sometimes used in architecture and engineering. 3-points seldom used …add little extra realism

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Perspective Projections - Cont.• 2-points sometimes used in architecture

and engineering

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Perspective projectionsPerspective projections

Perspective views and principal vanishing points of a cube for various orientations of the view plane relative to the principal axes of the object

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Perspective Projections - Cont.• 3-points seldom used … add little extra

realism

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Perspective projections

Perspective projection of objects at different distances from the view plane: the shorter segment s1 appears longer => no preservation of relative proportions

s1

s2

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Parallel Projections

• Parallel projections are defined by a Direction Of Projection (DOP) and a projection plane

• DOP also called projection vector

• Coordinate positions are transformed to the view plane along parallel lines

• Important property: they preserve relative proportions of objects

• Less realistic effect

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Parallel projections

DOP(projection plane)

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Parallel Projections• Classified as orthographic or oblique

• The DOP makes 2 angles with the projection plane

• Orthographic means DOP is perpendicular to the projection plane, i.e. both angles are 90 degrees

• Oblique means DOP not perpendicular– i.e. one or both angles are not equal to 90 degrees

DOP DOP

orthographic oblique

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Parallel projections

• 3 parallel projections of an object showing relative proportions from different viewing positions.

• Used in engineering and architectural drawings: object represented through a set of views => its appearance can be reconstructed from these views

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Orthographic projections

Model

Front Elevation

Side Elevation

Plan Elevation

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Orthographic projections

Orthographic projections of an object (plan + elevation views)

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ObliqueOblique projections are used to give a “flat” view of the most important side and view of two other sides at a given angle φ (e.g., 45 degrees)This view is most useful if we need to show a small detail on one side but don't need all other complete views (e.g., plan and elevation).

φ

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Perspective Projections - 1

• The larger-sized object will be smaller on the view-plane because it is further away from the Centre of Projection (COP)

COP

ViewPlane

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Perspective Projections - 2

• To obtain a perspective projection of a three-dimensional object, we transform points along projection lines that meet at the COP

• Example: z=d projection plane; (0,0,0) COP

z-axis

PP(xP,yP,d)

y-axis

d

P(x,y,z)x-axis

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Other Views• View along y

axis

z-axis

xP

x-axis

d

P(x,y,z)

Projection Plane

View along x axis

Projection Plane

d z-axis

y-axis

P(x,y,z)

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Perspective projections: equations• To calculate PP = (xP,yP,zP), the perspective

projection of (x,y,z) onto the projection plane at z = d we use the shared properties of similar triangles

=dxP

zx

z-axis

xP

x-axis

d

P(x,y,z)

Projection Plane

=dyP

zy

Mutiplying each side by d yields

=xPz

d.x = z/dx =yP

zd.y = z/d

y

The distance d is just a scale factor applied to xP and yP

All values of z are allowable except z = 0

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Homogeneous Matrix Form• Using a 3D homogeneous coordinate

representation, we can write the perspective projection transformation matrix form

d/z 0 0 0

0 d/z 0 0

0 0 d/z 0

0 0 1/z 0

=

Xp

Yp

Zp

1

x

y

z

1

z/dx ,(xP, yP, zP, 1) =

z/dy ,( )d, 1

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Projection Equations Cont.• This makes sense intuitively: the further

away from the origin (our COP) a point P is, the larger its z value

• By dividing the x and y co-ordinates of every point of an object by the z coordinate means that objects further away will have each x and y divided by a larger number, and therefore the projection onto the projection plane will be much smaller than objects that are closer to the COP (in this case the origin)